Function invariant and parameter scale-free transformation methods

The parameter matrices of factor analysis and principal component analysis are arbitrary with respect to the scale of the factors or components; typically, the scale is fixed so that the factors have unit variance. Oblique transformations to optimize an objective statement of a principle such as simple structure or factor simplicity yield arbitrary solutions, unless the criterion function is invariant with respect to the scale of the factors, or the parameter matrix is scale free with respect to the factors. Criterion functions that are factor scale-free have a number of invariance characteristics, such as being equally applicable to primary pattern or reference structure matrices. A scale-invariant simple structure function of previously studied function components is defined. First and second partial derivatives are obtained, and Newton-Raphson iterations are utilized. The resulting solutions are locally optimal and subjectively pleasing.Aspects of this paper were presented at the 1970 and 1974 annual meetings, Society of Multivariate Experimental Psychology, and the 1975 annual meeting, Psychometric Society. This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and research grants (MH24149 and DA01070) from the U. S. Public Health Service. The assistance of Bonnie Barron, Sik-Yum Lee, and several extremely helpful reviewers is gratefully acknowledged.